Nos quedaremos en el corral hasta que la granjera termine de dar de comer a los animales.

Because quedarse is the verb being used here, not plain quedar.

• quedarse = to stay / to remain
• quedar without the pronoun can mean other things, such as to remain, to be left, to meet, or to fit, depending on context

So nos quedaremos means we will stay. The nos matches we.

Because the staying happens after the moment of speaking. Spanish uses the simple future here just like English often uses will.

• nos quedaremos = we will stay

You could also hear nos vamos a quedar, which also means we are going to stay. Both are natural.

Because en shows location, while a / al shows movement toward a place.

• en el corral = in the pen / in the yard
• al corral = to the pen / to the yard

Here the sentence is about staying in a place, not going to it, so en is the right choice.

Corral usually means an enclosed area for animals, such as a pen, yard, or animal enclosure.

In this sentence, it most likely refers to the place on a farm where the animals are kept. The exact English word can vary with context, but the Spanish word corral is very common for that kind of space.

Granjera is the feminine form of granjero, so it means female farmer or farm woman.

It is used because the sentence is talking about a woman. Spanish adjectives and many nouns change form to match gender:

• el granjero = the male farmer
• la granjera = the female farmer

This word especially fits a farm context with animals.

Because after hasta que, Spanish uses the subjunctive when the action has not happened yet and is still in the future from the speaker’s point of view.

Here, the farmer has not finished yet. That is why Spanish says:

• hasta que la granjera termine

not

• hasta que la granjera termina

So termine is present subjunctive, even though the meaning is future.

That is a very common Spanish pattern. After time expressions like hasta que, cuando, or después de que, Spanish often uses the present subjunctive to talk about a future event that is pending.

So:

• hasta que termine = until she finishes

Even though it refers to the future, Spanish does not normally use the future tense there.

Because terminar de + infinitive is a fixed structure meaning to finish doing something.

So:

• terminar de comer = to finish eating
• terminar de limpiar = to finish cleaning
• terminar de dar de comer = to finish feeding

In this sentence, termine de dar de comer means finishes feeding.

Literally, dar de comer means to give to eat. In natural English, that becomes to feed.

It is a very common everyday Spanish expression.

• dar de comer a los animales = to feed the animals

You could also use alimentar, but dar de comer is often more natural in ordinary speech, especially when talking about people or animals being given food.

Because dar normally takes the pattern dar algo a alguien — to give something to someone.

In dar de comer a los animales, the animals are the receivers of the food, so they are introduced with a.

This is not exactly the same as the so-called personal a. It is mainly there because dar needs a recipient:

• dar comida a los animales
• dar de comer a los animales

Spanish uses definite articles more often than English does.

Here they sound natural because the sentence is talking about specific, identifiable things in the situation:

• el corral = the pen / the yard
• la granjera = the female farmer
• los animales = the animals

Even where English might sometimes leave an article out, Spanish often keeps it.

Sometimes yes, depending on context.

Spanish often uses the present to talk about near-future plans:

• Nos quedamos en el corral hasta que la granjera termine...

This can sound like We’re staying in the pen until...

But nos quedaremos is a clear, straightforward future form, and it fits perfectly if the speaker is stating what will happen.

Yes. It depends on whether the action is viewed as completed/factual or still pending.

• Future / not completed yet → subjunctive
  Nos quedaremos aquí hasta que la granjera termine.

• Past / completed → indicative
  Nos quedamos allí hasta que la granjera terminó.

So in your sentence, the finishing has not happened yet, which is why termine is correct.